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Theorem bcthlem2 24692
Description: Lemma for bcth 24696. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2 𝐽 = (MetOpenβ€˜π·)
bcthlem.4 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
bcthlem.5 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
bcthlem.6 (πœ‘ β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
bcthlem.7 (πœ‘ β†’ 𝑅 ∈ ℝ+)
bcthlem.8 (πœ‘ β†’ 𝐢 ∈ 𝑋)
bcthlem.9 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
bcthlem.10 (πœ‘ β†’ (π‘”β€˜1) = ⟨𝐢, π‘…βŸ©)
bcthlem.11 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
Assertion
Ref Expression
bcthlem2 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Distinct variable groups:   π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝐢,π‘Ÿ,π‘₯   𝑔,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧,𝐷   𝑔,𝐹,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝐽,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   𝑔,𝑀,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   πœ‘,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧   π‘₯,𝑅   𝑔,𝑋,π‘˜,𝑛,π‘Ÿ,π‘₯,𝑧
Allowed substitution hints:   πœ‘(𝑔)   𝐢(𝑧,𝑔,π‘˜,𝑛)   𝑅(𝑧,𝑔,π‘˜,𝑛,π‘Ÿ)

Proof of Theorem bcthlem2
StepHypRef Expression
1 bcthlem.11 . . . . 5 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)))
2 fvoveq1 7381 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘”β€˜(π‘˜ + 1)) = (π‘”β€˜(𝑛 + 1)))
3 id 22 . . . . . . . 8 (π‘˜ = 𝑛 β†’ π‘˜ = 𝑛)
4 fveq2 6843 . . . . . . . 8 (π‘˜ = 𝑛 β†’ (π‘”β€˜π‘˜) = (π‘”β€˜π‘›))
53, 4oveq12d 7376 . . . . . . 7 (π‘˜ = 𝑛 β†’ (π‘˜πΉ(π‘”β€˜π‘˜)) = (𝑛𝐹(π‘”β€˜π‘›)))
62, 5eleq12d 2832 . . . . . 6 (π‘˜ = 𝑛 β†’ ((π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ↔ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›))))
76rspccva 3581 . . . . 5 ((βˆ€π‘˜ ∈ β„• (π‘”β€˜(π‘˜ + 1)) ∈ (π‘˜πΉ(π‘”β€˜π‘˜)) ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
81, 7sylan 581 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)))
9 bcthlem.9 . . . . . 6 (πœ‘ β†’ 𝑔:β„•βŸΆ(𝑋 Γ— ℝ+))
109ffvelcdmda 7036 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))
11 bcth.2 . . . . . . 7 𝐽 = (MetOpenβ€˜π·)
12 bcthlem.4 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
13 bcthlem.5 . . . . . . 7 𝐹 = (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))})
1411, 12, 13bcthlem1 24691 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ β„• ∧ (π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+))) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
1514expr 458 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜π‘›) ∈ (𝑋 Γ— ℝ+) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))))
1610, 15mpd 15 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑛𝐹(π‘”β€˜π‘›)) ↔ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)))))
178, 16mpbid 231 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))))
18 cmetmet 24653 . . . . . . . . . . . 12 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
1912, 18syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))
20 metxmet 23690 . . . . . . . . . . 11 (𝐷 ∈ (Metβ€˜π‘‹) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2119, 20syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2211mopntop 23796 . . . . . . . . . 10 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2321, 22syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐽 ∈ Top)
24 xp1st 7954 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋)
25 xp2nd 7955 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ+)
2625rpxrd 12959 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)
2724, 26jca 513 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*))
28 blssm 23774 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
29283expb 1121 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ ((1st β€˜(π‘”β€˜(𝑛 + 1))) ∈ 𝑋 ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) ∈ ℝ*)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
3021, 27, 29syl2an 597 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† 𝑋)
31 df-ov 7361 . . . . . . . . . . . 12 ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
32 1st2nd2 7961 . . . . . . . . . . . . 13 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ (π‘”β€˜(𝑛 + 1)) = ⟨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩)
3332fveq2d 6847 . . . . . . . . . . . 12 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) = ((ballβ€˜π·)β€˜βŸ¨(1st β€˜(π‘”β€˜(𝑛 + 1))), (2nd β€˜(π‘”β€˜(𝑛 + 1)))⟩))
3431, 33eqtr4id 2796 . . . . . . . . . . 11 ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3534adantl 483 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((1st β€˜(π‘”β€˜(𝑛 + 1)))(ballβ€˜π·)(2nd β€˜(π‘”β€˜(𝑛 + 1)))) = ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))))
3611mopnuni 23797 . . . . . . . . . . . 12 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3721, 36syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑋 = βˆͺ 𝐽)
3837adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ 𝑋 = βˆͺ 𝐽)
3930, 35, 383sstr3d 3991 . . . . . . . . 9 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽)
40 eqid 2737 . . . . . . . . . 10 βˆͺ 𝐽 = βˆͺ 𝐽
4140sscls 22410 . . . . . . . . 9 ((𝐽 ∈ Top ∧ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† βˆͺ 𝐽) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
4223, 39, 41syl2an2r 684 . . . . . . . 8 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))))
43 difss2 4094 . . . . . . . 8 (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
44 sstr2 3952 . . . . . . . 8 (((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4542, 43, 44syl2im 40 . . . . . . 7 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4645a1d 25 . . . . . 6 ((πœ‘ ∧ (π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+)) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))))
4746ex 414 . . . . 5 (πœ‘ β†’ ((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) β†’ ((2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) β†’ (((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›)) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))))
48473impd 1349 . . . 4 (πœ‘ β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
4948adantr 482 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (((π‘”β€˜(𝑛 + 1)) ∈ (𝑋 Γ— ℝ+) ∧ (2nd β€˜(π‘”β€˜(𝑛 + 1))) < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1)))) βŠ† (((ballβ€˜π·)β€˜(π‘”β€˜π‘›)) βˆ– (π‘€β€˜π‘›))) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›))))
5017, 49mpd 15 . 2 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
5150ralrimiva 3144 1 (πœ‘ β†’ βˆ€π‘› ∈ β„• ((ballβ€˜π·)β€˜(π‘”β€˜(𝑛 + 1))) βŠ† ((ballβ€˜π·)β€˜(π‘”β€˜π‘›)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   βˆ– cdif 3908   βŠ† wss 3911  βŸ¨cop 4593  βˆͺ cuni 4866   class class class wbr 5106  {copab 5168   Γ— cxp 5632  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  1c1 11053   + caddc 11055  β„*cxr 11189   < clt 11190   / cdiv 11813  β„•cn 12154  β„+crp 12916  βˆžMetcxmet 20784  Metcmet 20785  ballcbl 20786  MetOpencmopn 20789  Topctop 22245  Clsdccld 22370  clsccl 22372  CMetccmet 24621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-n0 12415  df-z 12501  df-uz 12765  df-q 12875  df-rp 12917  df-xneg 13034  df-xadd 13035  df-xmul 13036  df-topgen 17326  df-psmet 20791  df-xmet 20792  df-met 20793  df-bl 20794  df-mopn 20795  df-top 22246  df-topon 22263  df-bases 22299  df-cld 22373  df-cls 22375  df-cmet 24624
This theorem is referenced by:  bcthlem3  24693  bcthlem4  24694
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